Locally Stationary UETC in Cosmology

• Locally Stationary UETC is a method that generalizes unequal-time correlators by imposing time-local stationarity to capture dynamic field correlations.
• The framework factorizes time dependence into mean and relative components, enabling accurate modeling of gravitational waves, CMB anisotropies, and thermalization in holography.
• It captures non-linear source decay through time-dependent kinetic energy factors, providing analytic templates for simulation calibration and observational forecasting.

A locally stationary unequal-time correlator (UETC) is a generalization of the unequal-time correlator concept used for modeling the temporal and spatial correlations of physical fields, allowing stationarity to be imposed not globally but locally in time. In this framework, the correlator depends explicitly on both the mean time (often denoted as $\eta_+ = (\eta_1 + \eta_2)/2$) and relative time ($\eta_- = \eta_1 - \eta_2$), capturing the time evolution of sources such as fluid motion, defects, or fields in dynamical and non-equilibrium settings. This formalism has been instrumental in the calculation of observables like gravitational wave (GW) spectra from cosmological phase transitions, cosmic microwave background (CMB) anisotropies sourced by defects, and diagnostics of thermalization in gauge/gravity duality.

1. Formal Definition and Mathematical Structure

The locally stationary UETC formalism splits the time arguments of the correlator into mean and relative components, $E(\eta_1, \eta_2, k) = E(\eta_+, \eta_-, k)$, and factorizes the time dependence such that it incorporates local stationarity. A representative example from cosmological GW modeling is:

$$E_\Pi(\eta_1, \eta_2, k) = 2 k^2 K^2(\eta_+) f(\eta_-, k)$$

where $K(\eta_+)$ is the time-dependent kinetic energy fraction (encoding the decay of fluid motion), and $f(\eta_-, k)$ represents a stationary correlator over the relative time, commonly inherited from sound-shell or defect models (Stomberg et al., 6 Aug 2025). This formalism is explicitly time-dependent, reflecting non-linear effects and source decay. In contrast, a globally stationary correlator would set $K(\eta_+)$ constant.

2. Physical Motivation and Applicability

Locally stationary UETCs were developed in response to the need for accurate modeling of physical sources with non-linear temporal evolution:

• Gravitational wave production: In cosmological phase transitions, the bulk motion of plasma induced by expanding broken-phase bubbles is initially well described by linear, non-decaying sound waves. However, after the phase transition, non-linearities cause the fluid’s kinetic energy to decay rapidly. The locally stationary UETC encodes this transition by promoting the kinetic-energy factor in the source correlator to become time-dependent (Stomberg et al., 6 Aug 2025).
• Thermalization in holography: Evolution of singularities in UETCs encodes the transition from non-equilibrium to equilibrium in the dual gauge theory description. As the system thermalizes, the number and structure of singularities change, providing a direct probe of decoherence and time-localized correlations (Erdmenger et al., 2012).
• Defect cosmologies: In the computation of CMB anisotropies sourced by cosmic strings or other defects, UETCs are essential tools. Analytic expressions for UETCs in string networks facilitate rapid and accurate calculations across all relevant time and scale ranges (Avgoustidis et al., 2012).

3. Modeling Source Decay and Non-Linear Evolution

The primary advance in locally stationary UETCs is their capacity to describe source decay:

• Non-linear fluid decay: After a phase transition, the kinetic energy $K(t)$ of the cosmic fluid can be empirically modeled as a power law, $K(t) = K_0 (t / t_0)^{-b}$, with $b \gtrsim 0$ the decay exponent and $t_0$ the transition completion time. The correlator incorporates this decay via the $K^2(\eta_+)$ term, and the impact on GW power is captured through the integrated kinetic factor

$$K_{\mathrm{int,exp}}^2 = H_*^{-1} \int_{\eta_*}^{\eta_{\mathrm{fin}}} \frac{K^2(\eta_+)}{\eta_+^2} \eta_+ d\eta_+$$

This integral defines a suppression factor for the GW spectrum depending on the decay properties (Stomberg et al., 6 Aug 2025).

• Transition to equilibrium: In holographic settings, the UETC's singular structure evolves during the collapse of a shell toward black hole formation, and the transition to a smoother correlator marks the attainment of thermal equilibrium and loss of time-localized memory (Erdmenger et al., 2012).

4. Observational and Computational Implications

Locally stationary UETC models enable precise predictions and efficient computational approaches for a range of cosmological observables:

• Gravitational wave spectra: Using locally stationary UETCs, the GW energy density spectrum is expressed as

$$\Omega_{\mathrm{GW}}(k) = 3 F_{\mathrm{GW}}^0\, \tilde{\Omega}_{\mathrm{GW}}\, K_{\mathrm{int,exp}}^2\, \mathcal{R}_*\, S(k \mathcal{R}_*)$$

where $F_{\mathrm{GW}}^0$ accounts for redshift, $\tilde{\Omega}_{\mathrm{GW}}$ is a numerically determined efficiency, $\mathcal{R}_*$ the characteristic lengthscale, and $S(k \mathcal{R}_*)$ a doubly broken power-law spectral function (Stomberg et al., 6 Aug 2025). This formalism is implemented in the Python package CosmoGW for experimental forecasts and parameter estimation.

• Defect-sourced CMB anisotropies: Analytic expressions for the UETC in unconnected segment models (USM) for cosmic strings allow rapid computation via eigen-decomposition, replacing ensemble averages over thousands of network simulations (Avgoustidis et al., 2012). This approach significantly accelerates MCMC exploration of string parameter space.
• Diagnostics of thermalization: The structure and disappearance of singularities in UETCs in time (and complex time) serve as diagnostics for decoherence in gauge theories and help track progression toward equilibrium (Erdmenger et al., 2012).

5. Dependence on Physical Parameters

The GW spectrum and other observables computed via locally stationary UETCs have explicit and often non-linear dependence on underlying physical parameters:

• Phase transition parameters: Wall velocity ($v_w$), transition strength ($\alpha$), nucleation rate ($\beta/H_*$), and source duration ($\delta \eta_{\mathrm{fin}}$) all modulate both the amplitude and spectral features of the GW signal. In strong transitions, rapid decay of fluid motion (large $b$) leads to saturation of GW amplitude, reducing sensitivity to the precise source duration (Stomberg et al., 6 Aug 2025).
• String network parameters: RMS velocity ($v$), correlation length ($\xi$), and wiggliness ($\alpha$) control the analytic form and scaling properties of defect UETCs (Avgoustidis et al., 2012).

The locally stationary UETC framework allows simulation results (e.g., Higgsless phase transition GW simulations) to be interpolated as analytic templates, facilitating experimental forecasts and data-driven parameter estimation.

6. Connections to Broader Unequal-Time Correlator Developments

• Projection formalism for large-scale structure: Technique refinement in unequal-time correlator projections, including field-level and correlator-level methods, has clarified that first-order corrections can arise even for single-tracer power spectra when full temporal displacement information is retained (Steele et al., 13 Feb 2025). These corrections, grounded in local stationarity and derivative structures over Fourier conjugate variables, can be significant in interpreting cross-bin and single-bin correlations in large-scale cosmological surveys.
• Beyond equilibrium/statistical stationarity: The locally stationary UETC generalizes notions from equilibrium theory and stationary processes, essential for modeling temporally evolving sources where traditional stationary approximations break down.

7. Prospects and Limitations

The locally stationary UETC formalism constitutes a central theoretical advance in modeling temporally evolving sources for cosmology and high-energy theory. It furnishes analytic templates for use in simulation calibration, forecasting, and interpretation. Remaining challenges include the rigorous extension of the approach to more complex, multi-component, and non-linear systems; full incorporation of hydrodynamic or relativistic corrections; and systematic quantification of error when translating physical decay laws into computationally tractable suppression factors. Continued refinement and simulation validation—such as those underpinning implementations in packages like CosmoGW—is crucial for reliable confrontation with observational data from next-generation GW detectors and precision CMB and large-scale structure surveys.